SYSTEM AND METHOD FOR HIGH EFFICIENCY ELECTROCHEMICAL DESALINATION
The present disclosure relates to a capacitive deionization (CDI) system for desalinating salt water. The system may have a capacitor formed by spaced apart first and second electrodes, which enable a fluid flow containing salt water to pass either between them or through them. An input electrical power source is configured to generate an electrical forcing signal between the two electrodes. The electrical forcing signal represents a periodic signal including at least one of voltage or current, and which can be represented as a Fourier series. One component of the Fourier series is a constant, and a second component of the Fourier series is a sinusoidal wave of non-zero frequency which has the highest amplitude of the additive components of the Fourier series. The amplitude of the sinusoidal wave component is between 0.85 and 1.25 times the amplitude of the periodic signal.
This application claims the benefit of U.S. Provisional Application No. 62/675,244, filed on May 23, 2018. The entire disclosure of the above application is incorporated herein by reference.
STATEMENT OF GOVERNMENT RIGHTSThe United States Government has rights in this invention pursuant to Contract No. DE-AC52-07NA27344 between the U.S. Department of Energy and Lawrence Livermore National Security, LLC, for the operation of Lawrence Livermore National Laboratory.
FIELDThe present disclosure relates to systems and methods for desalination, and more particularly to systems and methods for cyclic adsorption/desorption based on electrochemical desalination.
BACKGROUNDThis section provides background information related to the present disclosure which is not necessarily prior art.
With growing water shortages globally, roughly one quarter of all water desalination demand is for inland brackish water desalination technology. Conventional methods like reverse osmosis or thermal methods are highly energy inefficient for low salinity (e.g., brackish) water desalination. Capacitive deionization (“CDI”) is an emerging, highly energy efficient brackish water desalination technology.
Like many other multi-physics problems, CDI involves coupling of multiple time scales and phenomena. CDI salt removal dynamics are determined by the interplay between electrical charging/discharging, which depends on cell ionic and electrical resistances and capacitance, coupled with bulk mass transport. Moreover, CDI is inherently periodic because electrical charging and discharging forcing functions result in periodic salt removal and regeneration phases.
CDI performance can be evaluated using a recently proposed set of metrics (e.g., see Hawks et al. “Performance Metrics for the Objective Assessment of Capacitive Deionization Systems.” arXiv preprint arXiv. 1805.03247 (2018)). These performance metrics include average concentration reduction, volumetric energy consumption, and productivity for 50% water recovery. Owing to the multi-physics nature of CDI, the desalination performance can be affected dramatically by the particular choice of operating method. Most of the previous research on CDI operation has centered around the use of constant current (“CC”) and/or constant voltage (“CV”), and very little attention has been given to other possible operational schemes. CC operation has been shown to consume less energy compared to CV, given equal amount of salt removal (Choi. “Comparison of Constant Voltage (CV) and Constant Current (CC) Operation in the Membrane Capacitive Deionisation Process,” Desalination Water Treatment 56.4 (2015), 921-928. https://doi.org/10.1080/19443994.2014.942379; Kang et al. “Comparison of Salt Adsorption Capacity and Energy Consumption Between Constant Current and Constant Voltage Operation in Capacitive Deionization,” Desalination 352 (2014), 52-57. https://doi.org/10.1016/j.desa1.2014.08.009; Qu et al., “Energy Consumption Analysis of Constant Voltage and Constant Current Operations in Capacitive Deionization,” Desalination 400, (2016): 18-24. https://doi.org/10/1016/i.desal.2016.09.014). CC can also achieve a controllable quasi-steady state effluent concentration. Conversely, CV can achieve faster rates of desalination, albeit with a tradeoff in energy consumption. Recent research around operational schemes for CDI have proposed mixed CC-CV modes, variable flow rate, changing feed concentration, and variable forcing function periods. Generally, such research/studies can be characterized as ad-hoc operational strategies geared toward the improvement of one (or few) metrics at the cost of others.
The co-inventors of the present disclosure know of no work which combines a theoretical framework and accompanying validation experiments which explore generalized control waveform shapes for CDI. In other words, to the understanding of the co-inventors, previous studies to date have only explored ad hoc operational schemes such as square waves in applied current or voltage. A key step in developing favorable operation modes for CDI involves understanding the role of arbitrary periodic forcing functions (including frequency and wave shape) on the aforementioned desalination performance metrics.
In summary, previously developed desalination systems have been limited in terms of being able to achieve optimum operational efficiency. More specifically, such previously developed desalination systems have not been able to address the complexity of cyclic CDI operation, which can have arbitrarily variable flow rate and charging modes, in a manner that optimizes overall efficiency of the CDI process.
SUMMARYThis section provides a general summary of the disclosure, and is not a comprehensive disclosure of its full scope or all of its features.
In one aspect the present disclosure relates to a capacitive deionization (CDI) system for desalinating salt water. The system may comprise a capacitor formed by a first electrode and a second electrode spaced apart from the first electrode, which enables a fluid flow containing salt water to pass at least one of between the first and second electrodes, or through the first and second electrodes. The system may also include an input electrical power source configured to generate an electrical forcing signal between the first and second electrodes, with the electrical forcing signal representing a periodic signal including at least one of voltage or current. The periodic signal can be represented as a Fourier series, and one component of the Fourier series is a constant. A second component of the Fourier series is a sinusoidal wave of non-zero frequency which has the highest amplitude of the additive components of the Fourier series. The amplitude of the sinusoidal wave component is between 0.85 and 1.25 times the amplitude of the periodic signal.
In another aspect the present disclosure relates to a capacitive deionization (CDI) system for desalinating salt water. The system may comprise a capacitor formed by a first electrode and a second electrode spaced apart from the first electrode, which enables a fluid flow containing salt water to pass at least one of between the first and second electrodes, or through the first and second electrodes. The system may further include an input electrical power source which is configured to generate an input electrical forcing signal between the first and second electrodes, wherein the input electrical forcing signal represents a periodic signal including at least one of voltage or current. The periodic signal may have a fundamental frequency within a factor of 10 of a resonant frequency ωres given, where ωres is given by:
wherein τ is a flow time scale given by
τ=V/Q
wherein V is a fluid volume contained with the CDI cell and Q is a time-averaged volume flow rate through the CDI cell, and wherein RC is the product of the characteristic resistance R and capacitance C of the electrical response of the CDI cell.
In another aspect the present disclosure relates to a capacitive deionization (CDI) system for desalinating salt water. The system may comprise a capacitor formed by a first electrode and a second electrode spaced apart from the first electrode, which enables a fluid flow containing salt water to pass at least one of between the first and second electrodes, or through the first and second electrodes. An input electrical power source may be provided. The input electrical power source may be configured to generate an electrical forcing signal between the first and second electrodes. The electrical forcing signal may represent a periodic signal including at least one of voltage or current, wherein the periodic signal has a fundamental frequency within a factor of 10 of a resonant frequency ωres given, where ωres is given by:
wherein τ is a flow time scale given by
τ=V/Q
and wherein V is a fluid volume contained with the CDI cell and Q is a time-averaged volume flow rate through the CDI cell, and wherein RC is the product of the characteristic resistance R and capacitance C of the electrical response of the CDI cell, and wherein the periodic signal has a Fourier series as additive sinusoidal components, and wherein the amplitude of the sinusoidal wave of non-zero frequency with the highest amplitude of the additive components of the Fourier series is between 0.85 and 1.25 times the amplitude of the forcing periodic signal.
In still another aspect the present disclosure relates to a method for desalinating salt water. The method may comprise using a capacitor formed by a first electrode and a second electrode spaced apart from the first electrode, which enables a fluid flow containing salt water to pass at least one of between the first and second electrodes, or through the first and second electrodes. The method may further comprise using an input electrical power source to apply an electrical forcing signal between the electrodes, and controlling the electrical forcing signal to provide at least one of voltage or current as a periodic wave with a fundamental frequency plus a constant component. The method further may include controlling the electrical forcing signal so that a Fourier series representation of the periodic wave includes a non-zero frequency sinusoidal mode which has the highest amplitude of the additive components of the Fourier series with an amplitude between 0.85 and 1.25 times the amplitude of the forcing periodic signal.
Further areas of applicability will become apparent from the description provided herein. The description and specific examples in this summary are intended for purposes of illustration only and are not intended to limit the scope of the present disclosure.
The drawings described herein are for illustrative purposes only of selected embodiments and not all possible implementations, and are not intended to limit the scope of the present disclosure. In the drawings:
Corresponding reference numerals indicate corresponding parts throughout the several views of the drawings.
Example embodiments will now be described more fully with reference to the accompanying drawings.
The present disclosure relates to a new capacitive deionization (CDI) system and method for cyclic adsorption/desorption based electrochemical desalination. The system and method of the present disclosure recognizes that CDI performance metrics can vary widely with operating methods, and further that conventional CDI operating methods such as constant current (CC) and constant voltage (CV) show advantages in either energy or salt removal performance, but not both. The present CDI system and method addresses these challenges through the implementation of a sinusoidal forcing voltage (or a sinusoidal current). The system and method of the present disclosure uses a dynamic system modeling approach, and quantifies the frequency response (amplitude and phase) of CDI effluent concentration. Using a wide range of operating conditions, the present disclosure demonstrates that CDI can be modeled as a linear time invariant system. This model is validated with experiments and shows that a sinusoid voltage operation can simultaneously achieve high salt removal and strong energy performance, thus making it superior to other conventional operating methods. Based on the underlying coupled phenomena of electrical charge (and ionic) transfer with bulk advection in CDI, the present disclosure validates, by presenting experimental data, results and effectiveness of using sinusoidal voltage forcing functions to achieve resonance-type operation for CDI. Despite the complexities of the present system, an important relation for the resonant time scale is set forth: the resonant time period (frequency) is proportional (inversely proportional) to the geometric mean of the flow residence time and the electrical (RC) charging time. Operation at resonance implies the optimal balance between absolute amount of salt removed (in moles) and dilution (depending on the feed volume processed), thus resulting in the maximum average concentration reduction for the desalinated water.
The present disclosure further develops the above model to generalize the resonant time-scale operation, and to provide responses for square and triangular voltage waveforms as two specific examples. To this end, the present disclosure also presents a general tool that uses Fourier analysis to construct CDI effluent dynamics for arbitrary input waveforms. Using this tool, it can be shown that most of the salt removal (˜95%) for square and triangular voltage forcing waveforms is achieved by the fundamental Fourier (sinusoidal) mode. The frequency of higher Fourier modes precludes high flow efficiency for these modes, so these modes consume additional energy for minimal additional salt removed. This deficiency of higher frequency modes further highlights the advantage of DC-offset sinusoidal forcing for the presently disclosed system and method of CDI operation.
2. A Resonant CDI Operation
The present disclosure is focused around CDI desalination dynamics and the use of a sinusoidal forcing current or voltage. For simplicity and without significant loss of applicability, the electrical response of the CDI cell is treated as a simple, series, linear RC circuit with effective R and C values, as determined in the following section Section 3.2. A highly simplified example of such a CDI cell 10 is shown in
For the CDI cell 10 electrical circuit, we assume a DC-offset sinusoidal forcing voltage given by
V(t)=Vdc+ΔV sin(ωt), (1)
where Vdc is the constant DC component of applied voltage (typically >0 V for good performance), ΔV is the amplitude of the sinusoid voltage, and ω is the forcing frequency. Under dynamic steady state (DSS) such that the initial condition has sufficiently decayed as per the CDI system's natural response, current I in the electrical circuit may be expressed as:
The result of Equation (2) can be represented as
I(t)=ΔI sin(ωt+ϕIV) (3)
where the amplitude, and the phase of current with respect to voltage, are given by
respectively.
Further, the dynamics that govern effluent concentration reduction Δc via the mixed reactor model approximation can be described as,
where Δc=c0−c(t) represents an appropriate reduction of the feed concentration c0 at the effluent, Q is flow rate, F is Faraday's constant, τ (=∀/Q) is the flow residence time scale (∀ is the mixed reactor volume), and
Equivalently,
Δc(t)=Δcac sin(ωt+ϕcV) (6)
where
is the maximum change in effluent concentration, and
is the phase of Δc with respect to V. Further, the phase of Δc with respect to current I is given by ϕcI=−arctan(ωτ). Also, the average concentration reduction at the effluent is given by Δcavg=2Δcac/π, and water recovery is 50%. Note the absolute concentration difference Δcac depends on extensive (versus mass-specific, intensive) CDI cell properties such as R, C, and cell volume. Importantly, Δcac is also a function of operational parameters such as Q, voltage window, and forcing frequency ω. We find that the basic coupling of RC circuit dynamics and mixed reactor flow directly results in what we here will refer to as a “resonant frequency”, ωres. This frequency maximizes effluent concentration reduction Δcac in Equation (6) and is simply the inverse geometric mean of the respective circuit and flow time scales,
Furthermore, the maximum average concentration reduction Δcavg,res achieved at the resonant frequency may be given by,
With reference to
Lastly, it will be noted that the dynamic system analysis presented in this section can also be derived using a Laplace transform formulation involving transfer functions for the CDI system. For readers who may find this more intuitive or familiar, we provide such a formulation in Section S1 below. Somewhat surprisingly, the present work is the first to develop such transfer function formulation for practical operations using CDI.
3.1 CDI Cell Design
The present disclosure further involved fabricating and assembling a flow between (fbCDI) cell using the radial-flow architecture described in Biesheuvel and van der Wal. “Membrane Capacitive Deionization,” J. Memb. Sci. 346, (2010) 256-262. https://doi.org/10/1016/J.MEMSCI.2009.09.043, Hemmatifar et al. “Energy Breakdown in Capacitive Deionization,” Water Research 104, (2016) 303-311. https://doi.org/10.1016/J.WATRES.2016.08.020, and Ramachandran et al. “Self Similarities in Desalination Dynamics and Performance Using Capacitive Deionization,” Water Research 140 (2018), 323-334. https://doi.org/10.1016/j.watres.2018.04.042. Five pairs of activated carbon electrodes (with 5 cm diameter, 300 μm thickness, and total dry mass of 2.65 g were stacked between 5 cm diameter, 130 μm thick titanium sheets which acted as current collectors. Two 180 μm thick non-conductive polypropylene circular meshes (McMaster-Carr, Los Angeles, Calif.) were stacked between each electrode pair as spacers, with an estimated porosity of ˜59%. The spacers had a slightly larger (˜5 mm) diameter than the electrodes and current collectors to prevent electrical short circuits.
3.2 Experimental Methods and Extraction of Model Parameters
An experimental setup consisted of the fbCDI cell, a 3 L reservoir filled with 20 mM potassium chloride (KCl) solution which was circulated in a closed loop, a peristaltic pump, a flow-through conductivity sensor close to the cell outlet, and a sourcemeter. We estimate less than 1% change in reservoir concentration based on adsorption capacity of our cell, and thus approximate influent concentration as constant.
The resistance and capacitance of the cell were characterized using simple galvanostatic charging, and these estimates were corroborated by electrochemical impedance spectroscopy (EIS) and cyclic voltammetry measurements using a potentiostat/galvanostat (e.g., such as available from Gamry Instruments, Warminster, Pa.). See SI Section S2 for data. We estimated a differential cell capacitance of 33±1.8 F (equivalently ˜44 F/cm3 and 49 F/g) and an effective series resistance of 2.85±0.28 Ohms, resulting in a system RC time scale of ˜94 s. To determine the mixed reactor cell volume ∀, we used an exponential fit to the temporal response (open-circuit flush) of the cell as described in Hawks et al., 2018a and Ramachandran et al., 2018, and we estimated V of 2.1±0.2 ml. For simplicity, all of the forced (sinusoidal, triangular and square voltage) responses presented in this work are at a constant flowrate of 2.3 ml/min, corresponding to a residence time scale τ (=∀/Q) of ˜55 sec. Thus, the operational and system parameters described here result in a resonant frequency ƒres (=ωres/2π) value of 2.2 mHz (using Equation (7)), and a corresponding resonant time scale Tres (=1/ƒres) of 450 sec. The water recovery was 51-57% for all the cases presented here.
4.0 Results and Discussion
4.1 Cdi as a First Order Linear Time Invariant (Lti) Dynamic Process—Response to Sinusoid Voltage Forcing
The desalination dynamics associated with CDI from a “dynamical system modeling” viewpoint were also studied. To this end, we subjected the CDI cell with a constant flow rate and operated with a sinusoidal voltage forcing. Further, we constrained the voltage of operation within reasonable limits: sufficiently low peak voltage such that the Coulombic losses were small, and a voltage window such that EDL charge efficiency could be approximated by a constant value (Hawks et al. “Quantifying the Flow Efficiency in Constant-Current Capacitive Deionization,” Water Research 129 (2018), 327-336. https://doi.org/10.1016/j.watres.2017.11.025; Kim et al. “Enhanced Charge Efficiency and Reduced Energy Use in Capacitive Deionization by Increasing the Discharge Voltage,” Journal of Colloid Interface Science 446 (2015), 317-326. https://doi.org/10.1016/J.JCIS.2014.08.041; Ramachandran et al. “Self Similarities in Desalination Dynamics and Performance Using Capacitive Deionization,” Water Research 140 (2018), 323-334. https://doi.org/10.1016/j.watres. 2018.04.042).
We thus infer that the desalination dynamics using CDI can be modeled to a good approximation as a first order linear time invariant (LTI) system under the following conditions: (i) constant flowrate (with advection dominated transport), (ii) small variation in dynamic EDL charge efficiency such that it can be approximated by a constant value, and (iii) high Coulombic efficiency (close to unity). LTI systems have well-developed tools for system analysis and control, and thus can be applied to analyzing CDI systems. In Section S3 below, we provide one anecdotal “off-design” sinusoidal input operation of CDI which results in significant distortion of the output concentration. Namely, we show the case of a large variation in EDL charge efficiency due to a large voltage window wherein effluent concentration exhibits a significant deviation from a sine wave. Further study of such deviations from linearity may be helpful in gaining a further understanding of variations in EDL charge efficiency.
Importantly, the predictions and experimental data of
Lastly, it should be noted that, although the focus herein is on sinusoidal voltage forcing functions, our work with the present CDI model indicates that sinusoidal current may also be used to characterize CDI dynamics. It is expected that sinusoidal applied current can also yield sinusoidal time variation of effluent concentration, thus extending the utility of the teachings presented herein. Preliminary experiments have been performed toward such a study and during such work it was observed that sinusoidal forcing currents easily lead to deviations from ideal behavior (and the model) due to unwanted Faradaic (parasitic) reactions. This results in an attenuation of concentration reduction in regions of high voltage, and a more complex natural response relaxation from the initial condition. Such sinusoidal forcing also preferably requires non-zero DC values for applied current to account for unavoidable Faradaic losses. Accordingly, it will be understood that a sinusoidal voltage will virtually always be preferred over sinusoidal current forcing as a more controllable and practical operating method.
4.2 Frequency Response: Bode Plot and Resonant Frequency Analysis for CDI
In this section, we present a frequency analysis of the response of current and effluent concentration in CDI for a forcing sinusoidal voltage.
4.2.1 Current Response
From
It will be noted in
4.2.2 Effluent Concentration Response
We here follow an averaging procedure similar to that of Section 4.2.1 to evaluate the phase and amplitude of the effluent response. For the effluent response, the only fitting parameter for the model is the product
Operation at the resonant frequency results in the maximum desalination depth Δcavg for a given voltage window. This is clearly supported by experiments and model results shown in
4.2.3 Physical Significance of the Resonant Frequency and Operation: Limiting Regimes
CDI as a practical dynamic process most often involves two dominant and independent time scales: (i) an RC time (electronic time scale associated with electrical circuit properties of the CDI system), and (ii) flow residence time (ionic transport time scale in a mixed reactor volume). The interplay between these two time scales determines the desalination depth Δcavg at the effluent. To better understand this interplay, we here describe operating scenarios corresponding to very high and very low operating frequencies.
At high frequency operation (ƒ>>ƒτ and ƒRC) the rapid forcing results in repeated desalination and regeneration (salt uptake and rejection) from and to approximately the same volume of water contained in the CDI cell. Further, the RC-type electrical response of the cell is such that high frequencies incompletely charge the capacitive elements of the cell. This wasteful operation consumes energy and leads to low Δcavg. For very low frequencies (ƒ<<ƒτ and ƒRC) or equivalently long cycle durations, the EDLs are fully charged (high EDL charge efficiency) and freshwater recovery at the effluent is high (flow efficiency close to unity; c.f. Section 4.3.2); each of which is favorable. However, in this limiting regime, the system can be characterized as suffering from the mitigating effect of “overly dilute” effluent. That is, after EDL charging, the majority of the charging phase is spent flushing feed water through (and out of) the cell. Similarly, after EDL discharge, the majority of the discharging phase is again spent flowing feed water. Both of these phases hence exhibit a low value of the inherently time-averaged magnitude of Δcavg.
A corollary to the discussion above is that, for a given CDI cell and flowrate, there exist two frequencies (ƒlow,Δc and ƒhigh,Δc) for which Δcavg in a cycle is the same (see
4.3 Energy Consumption and Charge Efficiency Depend Strongly on Operating Frequency
4.3.1 Energy Consumption
The frequency dependence of the volumetric energy consumption Ev (assuming 100% electrical energy recovery during discharge) may be defined as
These analyses show that the fundamental frequency of the input forcing function ƒ should be within a factor of 10 of ƒres. Here, “fundamental” frequency is the frequency at which the periodic signal repeats itself. That is, that the non-dimensional frequency ƒ/fres should vary within about 0.1 to 10. Ideally, and for best performance, the fundamental frequency of the input forcing function ƒ should be within a factor of 5 of ƒres (i.e. values of ƒ/fres within 0.2 to 5). Further, to account for salt removal in addition to the corresponding energy consumption, the inset graph of
ENAS is a measure of salt removed (in moles) per energy consumed (in Joules) per cycle. As frequency decreases, ENAS increases, reaches a maximum and then decreases. Importantly, note that the maximum ENAS occurs at a frequency close to (slightly less than) the resonant frequency ƒres, thus again highlighting the importance of operation near the resonant frequency for good overall CDI performance. We attribute the decrease in ENAS at low frequencies to Faradaic energy losses which can become a significant source of energy loss for long cycles.
Lastly, we note that our estimate for the volumetric energy consumption Ev in Equation (9) and
4.3.2 Charge Efficiency
The frequency dependence of the conversion of electrical input charge to ions removed, as calculated from the effluent stream, was also studied. This conversion may be quantified by defining the cycle charge efficiency as
Previous studies have shown that the cycle charge efficiency Λcycle can be expressed as a product of three efficiencies as Λcycle=λdlλcλfl=
To support our hypothesis, we developed the following analytical expression for flow efficiency λfl for a sinusoid voltage operation:
The associated derivation is given in the following Section S1. We compared the predicted flow efficiency versus frequency based on Equation (12) with the corresponding extracted values for flow efficiency values from experimental data (λfl=Λcycle/
4.4 Generalization of Resonant Frequency Operation for Other Conventional Operations (Square and Triangular Voltage Waveforms)
We here generalize the resonant frequency operation for other conventional forcing waveforms such as square voltage (typically referred to as constant voltage operation in CDI) and triangular voltage (an operation similar to constant current operation). We operated the CDI cell with square and triangular voltage waveforms at varying cycle frequencies between 0.7 to 1.1 V (see inset of
The data of
Of the three waveforms considered here, the square voltage waveform (CV) results in the highest Δcavg, followed by sinusoidal (less than square wave by ˜15%), and then triangular (less than square wave by ˜43%) voltage waveforms. However, the volumetric energy consumption Ev for the triangular voltage wave operation is the lowest, followed by sinusoidal (around 1.5× of the triangular waveform Ev), and then square (around 4× of the triangular waveform Ev) voltage waveforms (see
Together, the data of
4.5 Constructing Effluent Response for Arbitrary Forcing Functions
This section summarizes a Fourier analysis which is helpful in rationalizing the various merits of CDI control schemes. Without loss of generality, we will assume that periodic forcing of the CDI cell is controlled by voltage, although a similar approach can be developed for a current forcing. Equation (5) in Section 2 is the expression for the effluent response for a sinusoidal forcing voltage with frequency ω (=2πƒ=2π/T). Any arbitrary voltage forcing V(t) which is periodic with time period T (and phase of zero at t=0) can be decomposed into its Fourier series as
with Fourier coefficients an and bn given by
Each of the term in the summation in Equation (13) corresponds to a Fourier mode. As shown in Sections 2 and 4.1, CDI can be modeled accurately as a linear time invariant system (under appropriate operating conditions), thus obeying linear superposition of effluent responses due to multiple forcing functions. We thus here hypothesize that the generalized forced response for an arbitrary forcing function in Equation (13) can be obtained using linear superimposition of responses of its Fourier components (modes). Section 2 presented the frequency response of CDI for a single sine wave and we can now interpret that response as the response of any one of an arbitrary number of Fourier modes.
We here analyze two special cases of Equations (13)-(15) corresponding to square and triangular voltage forcing waveforms (as shown in the inset of
Note that for the triangular wave Fourier modes in Equation (17), the amplitudes of harmonics decay as 1/(2n−1)2, compared to the 1/(2n−1) decay for the square waveform (Equation (16)).
From
These observations lead us to conclude the shape of the input forcing function should be close to the fundamental sine wave (n=1) and that this periodic wave should be added to a time-averaged DC component. To this end, we here define the “amplitude” of any periodic signal to be the maximum absolute value difference between the value of the periodic signal and the time-averaged value (during one cycle) of the periodic signal. Given the aforementioned increase in energy associated with higher harmonics, we estimate that the ratio of the amplitude of the highest magnitude sinusoidal mode (n≥1) of the Fourier series of the input signal to the amplitude of the input signal should be between 0.85 and 1.25. Ideally, the ratio of the amplitude of the highest magnitude sinusoidal mode of a Fourier series representation of the input signal to the amplitude of the input signal should be between 0.9 and 1.1.
For both the square and triangular waves, approximately 95% of the Δcavg is achieved by the fundamental (sinusoidal) Fourier mode alone. Adding higher frequency modes therefore provides only a slight increase (or sometimes even a decrease) in salt removal as compared to the fundamental mode alone, but at the great cost of significant energy consumption. This analysis leads us to the hypothesis that, for constant flow and appropriately voltage thresholded operation of CDI, the sinusoidal voltage operation introduced here is likely a near ideal tradeoff between salt removal performance and energy consumption.
SUMMARY AND CONCLUSIONSThe present disclosure thus teaches a model based on a dynamic system approach for a CDI system. The analysis set forth herein considers the coupled effects of electrical circuit response and the salt transport dynamics of a CDI cell. The teachings presented herein show that CDI cells with properly designed voltage windows exhibit first-order and near-linear dynamical system response. Experiments were performed to validate the model, and both theory and the experiments were used to study CDI performance for a variety of operational regimes. For the first time, the present disclosure identifies an inherent resonant operating frequency for CDI equal to the inverse geometric mean of the RC and flow time scales of the cell. The present disclosure also quantifies the frequency-dependent amplitude and phase of the current and effluent concentration responses for a sinusoidal voltage forcing. The teachings presented herein show that CDI operation near resonant frequency enables maximum desalination depth Δcavg.
The present disclosure further demonstrates that resonant frequency operation can be generalized to other operation methods, and presented analysis of square and triangular voltage forcing waveforms as two relevant case studies. Based on our validated theory, we developed a generalized tool that utilizes Fourier analysis for constructing effluent response for arbitrary input forcing current/voltage waveforms for predicting CDI effluent response. The present disclosure strongly suggests that a sinusoidal forcing voltage for CDI is the ideal operational mode to balance the tradeoff of energy consumption and salt removal in constant flow operation.
S.1 Supplemental Theory for Sinusoidal Voltage/Current Forcing for CKI
The following provides further details around the theory for predicting desalination dynamics associated with a sinusoidal voltage with a direct current (DC) offset as a forcing for capacitive deionization (CDI) as presented in Section 2 above. We assume that the electrical response of CDI can be described to a good approximation by a linear series resistor-capacitor (RC) circuit 200 such as shown in
To describe salt removal and freshwater recovery at the effluent, we assume a continuously stirred tank model. We present our derivation below of the coupled dynamics in two parts. First, we solve for the RC circuit current response for a sinusoidal voltage forcing. Second, we solve for dynamics associated with the effluent concentration reduction using the solution from the previous step, and assuming a well-mixed reactor.
S1.1 RC Circuit Analysis
Assume a series RC circuit with a DC-offset sinusoidal forcing voltage given by
V(t)=Vdc+ΔV sin(ωt), (18)
where Vdc is the constant DC component of applied voltage, ΔV is the amplitude of the sinusoid voltage and ω is the forcing frequency. Denoting the capacitive voltage drop by Vc, Kirchhoff's voltage law applied to the circuit in
Equation (19) can be written as,
where {tilde over (V)}c=Vc−Vdc. For long-duration dynamic steady state operation such that the transient associated with natural response (due to non-zero initial conditions) has decayed, the solution to Equation (20) is described the particular solution. The particular solution to Equation (20) is,
Since the current given by
we obtain the current in the circuit from Equation (21) as
The result in equation (23) can be expressed as
I(t)=ΔI sin(ωt+ϕIV), (24)
where the current amplitude
and the phase of current with respect to voltage is given by
S1.2 Mixed Reactor Model
We use a continuously stirred tank reactor model for predicting the effluent concentration dynamics. In a mixed reactor model, the salt removal dynamics is given by
where τ is the flow residence time, and we have assumed constant dynamic charge efficiency,
Combining Equation (24) in (25), we derive
The solution to Equation (26) is
which can be simplified as
Equivalently,
Δc=Δcac sin(ωt+ϕcV), (29)
where
is the maximum change in effluent concentration, and the phase of Δc with respect to the forcing voltage V is given by
The phase of Δc with respect to current is given by ϕcl=ϕcV−ϕIV=−arctan(ωτ).
Note further that Δcac and ΔI are related by,
S1.3 Flow Efficiency for Sinusoidal Forcing
The number of moles of salt ΔN removed per cycle is given by
In addition, the charge transferred Δq to the CDI cell per cycle is given by
The cycle charge efficiency Λcycle (measure of moles of salt removed as calculated at the effluent to the electrical charge input in moles) is related to the flow efficiency λfl (measure of fresh water recovery at the effluent) through the following relation,
where Equations (31) and (32) have been used for the last equality in Equation (33). Substituting Equation (30) in (33), we thus obtain the expression for flow efficiency for the sinusoidal operation as
S1.4 Transfer Functions for CDI
In this section transfer functions are developed relating the output (effluent concentration reduction) to input (current or voltage) for dynamic steady state CDI operation, under appropriate conditions as mentioned in Section 4.1 above.
Applying a Laplace transform to Equation (19), the transfer function relating the capacitive voltage Vc to the applied voltage V is derived as,
where s is the Laplace variable (Laplace frequency domain).
Further, from Equation (22) we have
I(s)=sCVc(s). (36)
Using Equation (36) in (35), we obtain the transfer function relating the current in the CDI circuit and applied voltage as
Next, from the mixed reaction model (Equation 25), the transfer function relating the effluent concentration reduction to current can be obtained as
Combining Equations (38) and (37), we obtain the following transfer function relating the effluent concentration reduction and the applied voltage:
Equations (37)-(39) are the transfer functions that relate the input (current or voltage) to the output (effluent concentration reduction) for a linear time invariant CDI system.
S2. Cell Resistance and Capacitance Measurements
A series of preliminary experiments were performed to characterize the CDI cell resistance and capacitance. First, we used simple galvanostatic charging and discharging (see
where |ΔV|I→−I is the voltage drop when current reverses sign (with the same magnitude). For the cases presented in
To corroborate the cell resistance estimate, we performed electrochemical impedance spectroscopy (EIS) of the entire assembled cell with 20 mM KCl solution and at flow rate of 2.3 ml/min. For EIS measurements (see
To verify the cell capacitance estimate, we performed cyclic voltammetry for the entire cell. For cyclic voltammetry, we used a scan rate of 0.2 mV/s, flow rate of 2.3 ml/min, and 20 mM KCl solution, and performed measurements till a steady state was reached. In
S3. Example of an Off-Design Sinusoidal Operation
To illustrate an operation wherein the effluent concentration variation with time is not sinusoidal for a DC-offset sinusoidal voltage forcing, we show in
S4. Coulombic Efficiency for Sinusoidal Operation
In this section we present the Coulombic efficiency data for sinusoidal voltage operation between 0.6 to 1.0 V, and 0.7 to 1.1 V, as a supplement to the data presented in
S5. Measured Effluent Concentration and Current Data for Square and Triangular Voltage Forcing Waveforms at Various Frequencies
S6. Volumetric Energy Consumption and ENAS with No Energy Recovery During Discharge
In this section a study is presented of the energy consumption metrics (volumetric energy consumption and energy normalized adsorbed salt ENAS) assuming 0% energy recovery during discharge. The volumetric energy consumption with 0% energy recovery Ev and the corresponding energy normalized adsorbed salt (ENAS) are defined as
In
Furthermore, the experimental data of the inset graph of
The foregoing description of the embodiments has been provided for purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure. Individual elements or features of a particular embodiment are generally not limited to that particular embodiment, but, where applicable, are interchangeable and can be used in a selected embodiment, even if not specifically shown or described. The same may also be varied in many ways. Such variations are not to be regarded as a departure from the disclosure, and all such modifications are intended to be included within the scope of the disclosure.
Example embodiments are provided so that this disclosure will be thorough, and will fully convey the scope to those who are skilled in the art. Numerous specific details are set forth such as examples of specific components, devices, and methods, to provide a thorough understanding of embodiments of the present disclosure. It will be apparent to those skilled in the art that specific details need not be employed, that example embodiments may be embodied in many different forms and that neither should be construed to limit the scope of the disclosure. In some example embodiments, well-known processes, well-known device structures, and well-known technologies are not described in detail.
The terminology used herein is for the purpose of describing particular example embodiments only and is not intended to be limiting. As used herein, the singular forms “a,” “an,” and “the” may be intended to include the plural forms as well, unless the context clearly indicates otherwise. The terms “comprises,” “comprising,” “including,” and “having,” are inclusive and therefore specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. The method steps, processes, and operations described herein are not to be construed as necessarily requiring their performance in the particular order discussed or illustrated, unless specifically identified as an order of performance. It is also to be understood that additional or alternative steps may be employed.
When an element or layer is referred to as being “on,” “engaged to,” “connected to,” or “coupled to” another element or layer, it may be directly on, engaged, connected or coupled to the other element or layer, or intervening elements or layers may be present. In contrast, when an element is referred to as being “directly on,” “directly engaged to,” “directly connected to,” or “directly coupled to” another element or layer, there may be no intervening elements or layers present. Other words used to describe the relationship between elements should be interpreted in a like fashion (e.g., “between” versus “directly between,” “adjacent” versus “directly adjacent,” etc.). As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items.
Although the terms first, second, third, etc. may be used herein to describe various elements, components, regions, layers and/or sections, these elements, components, regions, layers and/or sections should not be limited by these terms. These terms may be only used to distinguish one element, component, region, layer or section from another region, layer or section. Terms such as “first,” “second,” and other numerical terms when used herein do not imply a sequence or order unless clearly indicated by the context. Thus, a first element, component, region, layer or section discussed below could be termed a second element, component, region, layer or section without departing from the teachings of the example embodiments.
Spatially relative terms, such as “inner,” “outer,” “beneath,” “below,” “lower,” “above,” “upper,” and the like, may be used herein for ease of description to describe one element or feature's relationship to another element(s) or feature(s) as illustrated in the figures. Spatially relative terms may be intended to encompass different orientations of the device in use or operation in addition to the orientation depicted in the figures. For example, if the device in the figures is turned over, elements described as “below” or “beneath” other elements or features would then be oriented “above” the other elements or features. Thus, the example term “below” can encompass both an orientation of above and below. The device may be otherwise oriented (rotated 90 degrees or at other orientations) and the spatially relative descriptors used herein interpreted accordingly.
Claims
1. A capacitive deionization (CDI) system for desalinating salt water, the system comprising:
- a capacitor formed by a first electrode and a second electrode spaced apart from the first electrode, which enables a fluid flow containing salt water to pass at least one of between the first and second electrodes, or through the first and second electrodes;
- an input electrical power source;
- the input electrical power source configured to generate an electrical forcing signal between the first and second electrodes;
- the electrical forcing signal representing a periodic signal including at least one of voltage or current;
- wherein the periodic signal can be represented as a Fourier series;
- wherein one component of the Fourier series is a constant;
- wherein a second component of the Fourier series is a sinusoidal wave of non-zero frequency which has the highest amplitude of the additive components of the Fourier series; and
- wherein the amplitude of said sinusoidal wave component is between 0.85 and 1.25 times the amplitude of the periodic signal.
2. The system of claim 1 wherein the amplitude of said sinusoidal wave component is between 0.9 and 1.2 times the amplitude of the forcing periodic signal.
3. The system of claim 1, further comprising an electronic control system in communication with the input electrical power source for controlling the input electrical power source.
4. The system of claim 3, further comprising an electrical conductivity meter for measuring an electrical conductivity of effluent concentration of the salt water, and providing a signal representative of same back to the electronic control system.
5. The system of claim 1, wherein the electrical forcing signal comprises a DC-offset sinusoidal forcing voltage represented by:
- V(t)=Vdc+ΔV sin(ωt)+Vc(t)
- where Vc(t) is a component whose RMS value is less than 0.5 of the RMS value of Vdc and ΔV.
6. The system of claim 1, wherein the frequency of the electrical forcing signal is a value determined by the characteristic flow residence time of fluid in a mixed reactor volume and the characteristic resistance and capacitance electrical response of the system.
7. The system of claim 1, wherein the cycling frequency of the electrical forcing signal is within a factor of 10 of the resonant frequency (ωres) of the system determined by ω res = 1 τ RC where R is the characteristic electrochemical resistance of the system, C is ionic capacitance of the system, and τ is a flow time scale defined by: where V is a fluid volume contained with the CDI cell and Q is a time-averaged volume flow rate through the CDI cell.
- τ=V/Q
8. The system of claim 7, wherein the cycling frequency of the electrical forcing signal is within a factor of 5 of the resonant frequency (ωres).
9. The system of claim 1, further comprising first and second current collectors coupled to the first and second electrodes, respectively, for receiving the electrical forcing signal; and
- wherein the first and second electrodes comprise porous carbon electrodes.
10. A capacitive deionization (CDI) system for desalinating salt water, the system comprising: ω res = 1 τ RC
- a capacitor formed by a first electrode and a second electrode spaced apart from the first electrode, which enables a fluid flow containing salt water to pass at least one of between the first and second electrodes, or through the first and second electrodes;
- an input electrical power source;
- the input electrical power source configured to generate an input electrical forcing signal between the first and second electrodes;
- the input electrical forcing signal representing a periodic signal including at least one of voltage or current; the periodic signal having a fundamental frequency within a factor of 10 of a resonant frequency ωres given, where ωres is given by:
- wherein τ is a flow time scale given by τ=V/Q
- wherein V is a fluid volume contained with the CDI cell and Q is a time-averaged volume flow rate through the CDI cell; and
- wherein RC is the product of the characteristic resistance R and capacitance C of the electrical response of the CDI cell.
11. The system of claim 10, wherein the periodic signal has a fundamental frequency within a factor of 5 of a resonant frequency ωres.
12. The system of claim 10, wherein the periodic signal has a fundamental frequency within a factor of 2 of a resonant frequency ωres.
13. The system of claim 10, further comprising first and second current collectors coupled to the first and second electrodes, respectively, for receiving the electrical forcing signal; and
- wherein the first and second electrodes comprise porous carbon electrodes.
14. A capacitive deionization (CDI) system for desalinating salt water, the system comprising: ω res = 1 τ RC
- a capacitor formed by a first electrode and a second electrode spaced apart from the first electrode, which enables a fluid flow containing salt water to pass at least one of between the first and second electrodes, or through the first and second electrodes;
- an input electrical power source;
- the input electrical power source configured to generate an electrical forcing signal between the first and second electrodes;
- the electrical forcing signal representing a periodic signal including at least one of voltage or current;
- wherein the periodic signal has a fundamental frequency within a factor of 10 of a resonant frequency ωres given, where ωres is given by:
- wherein τ is a flow time scale given by τ=V/Q wherein V is a fluid volume contained with the CDI cell and Q is a time-averaged volume flow rate through the CDI cell; wherein RC is the product of the characteristic resistance R and capacitance C of the electrical response of the CDI cell;
- wherein the periodic signal has a Fourier series as additive sinusoidal components; and
- wherein the amplitude of the sinusoidal wave of non-zero frequency with the highest amplitude of the additive components of the Fourier series is between 0.85 and 1.25 times the amplitude of the forcing periodic signal.
15. The system of claim 14, further comprising first and second current collectors coupled to the first and second electrodes, respectively, for receiving the electrical forcing signal; and
- wherein the first and second electrodes comprise porous carbon electrodes.
16. The system of claim 14, wherein the periodic signal has a fundamental frequency within a factor of 10 of a resonant frequency ωres and wherein the highest amplitude of the additive components of the Fourier series is between 0.9 and 1.2 times the amplitude of the forcing periodic signal.
17. A method for desalinating salt water, the method comprising:
- using a capacitor formed by a first electrode and a second electrode spaced apart from the first electrode, which enables a fluid flow containing salt water to pass at least one of between the first and second electrodes, or through the first and second electrodes;
- using an input electrical power source to apply an electrical forcing signal between the electrodes;
- controlling the electrical forcing signal to provide at least one of voltage or current as a periodic wave with a fundamental frequency plus a constant component; and
- further controlling the electrical forcing signal so that a Fourier series representation of the periodic wave includes a non-zero frequency sinusoidal mode which has the highest amplitude of the additive components of the Fourier series with an amplitude between 0.85 and 1.25 times the amplitude of the forcing periodic signal.
18. The method of claim 17, further comprising controlling the electrical forcing signal so that the Fourier series representation of the periodic wave includes a sinusoidal wave with a finite frequency and which has an amplitude that is between 0.9 and 1.2 times the amplitude of the forcing periodic signal.
Type: Application
Filed: May 21, 2019
Publication Date: Nov 28, 2019
Patent Grant number: 10875792
Inventors: Steven HAWKS (Livermore, CA), Michael STADERMANN (Pleasanton, CA), Juan G. SANTIAGO (Stanford, CA), Ashwin RAMACHANDRAN (Stanford, CA)
Application Number: 16/418,487